Mind the gap and the map: Measuring distributional and spatial income polarization

Getis-Ord Lecture in Spatial Analysis

Sergio J. Rey

srey@sdsu.edu

Center for Open Geographical Science

Department of Geography

San Diego State University

February 13, 2025

I would like to take a few moments to share with you the impact these two scholars have had on my career.

I first met arthur in 1986 when I gave my first ever presentation at the AAG meeting. I don’t remember anything I said during my presentation, as I was terrified as Art and Mike Goodchild were sitting in the first row only a few feet from me.

I do remember, however, that after the session, Art came up and introduced himself and shared, in his gentle but serious manner a number of constructive suggestions. At that point I had no idea I would be lucky enough to be his colleague at SDSU only a few years later, or that I would benefit enormously from his counsel and support for over 36 years.

One of the many positive externalities you experience when Art is in your Department is a seemingly endless number of visits from predominate spatial scientistis and scholars. One such person was Keith Ord, who visited the first year I started as an assistant professor at SDSU in 1992. Keith met with me for an hour and the entire conversion consisted of Keith asking me questions about my research and offering invaluable encouragement on what I was doing and could do.

That is just me personally, benefiting from their gifts.

If we think of what they have given spatial analysis and many other disciplines, their collaboration has truly been transformative.

Their collaboration has resulted in two of the five most cited papers in GA. That represents impacts not just in the field of spatial analysis, but the wide scope of the applications of their methods across disciplines.

They put local analysis on the map.

I would like to thank the Getis-Ord committee for selecting me for this honor.

Overview

Polarization
Framework
Empirical Illustration
Conclusion

This paper presents a family of spatial polarization indices that integrates distributional and spatial dimensions to measure economic polarization. Traditional polarization metrics either focus solely on distributional aspects or rely on exogenously defined regional partitions, limiting their applicability to dynamic spatial economies. Building on economic polarization theory and spatial inequality decomposition, the proposed indices incorporate spatial adjacency relations and dynamically evolving regional configurations. By constructing intersection graphs that capture both attribute similarity (e.g., income levels) and geographic associations, this framework identifies spatially coherent clusters, and multiscalar measures of spatial polarization across global, meso, and local scales. The indices extend beyond bipolarization to multipolarization.

Polarization

Context

Regional Science Literature

Revenge Narrative

Rodríguez-Pose (2018)

Partisan Sorting

Revenge Realized

Rey and Casimiro Vieyra (2023)

Long-standing spatial inequalities within national systems have been identified as leading to a rise in political populism with strong territorial as opposed to social foundations (Rodríguez-Pose, 2018). The recent history of Mexico serves as an exemplar of this dynamic. In 2018, for the second time in history, a political party other than the PRI (Institutional Revolutionary Party) had won the Mexican Presidential election. Andres Manuel Lopez Obrador (AMLO) accomplished what he had initially envisioned in 2006, taking away power from corrupt political parties and institutions. AMLO’s victory was no small feat. With his newly formed political party National Regeneration Movement (MORENA), he had accomplished a landslide victory over rival political parties PRI and PAN by winning thirty-one out of the thirty-two Mexican states. Fig. 1 shows the presidential election results from 2012 to 2018. The staggering contrast in the results shows how AMLO leveraged the outrage from those who had suffered the most. His victory was backed by the social unrest born out of increased corruption, poverty, and violence that had plagued the country.

Spatial Polarization

Consensus

Important to address

Uncertainty (Pike et al. 2024)

How to measure?
What index?
What scale?
How to treat dynamics?

Dynamics and Polarization: Separated at Birth?

Quah (1996)

Wolfson (1994)

\[\begin{equation} P_W = (2G_B - G) \frac{\mu}{m}. \end{equation}\] where \(G_B\) is the Gini index of inequality between the two halves, \(G\) is the Gini index for the entire distribution, \(\mu\) is the mean of the distribution, and \(m\) is the median. Letting \(L(p)\) be the ordinate of the Lorenz curve at the \(p_{th}\) percentile, the discrepancy between perfect equality and the Lorenz curve at the median is: \(D_{50} = 0.5 - L(0.5)\). Given that \(G_B = D_{50}\), the Gini index can be rewritten as \(G=G_B + G_W\) with \(G_W\) representing inequality within the two halves of the distribution. This implies the polarization index is: \[\begin{equation} P_W = (G_B-G_W)\frac{\mu}{m}. \end{equation}\]

Bipolarization

Components

Alienation: Increasing gap between the two halves
Group Identification: decrease in local variance

Bipolarization vs. Inequality

Not synonyms
Pigou-Dalton: progressive income transfer must reduce inequality
Same transfer may increase polarization (e.g., Musk to Trump)

Multipolarization (Esteban and Ray 1994)

Division into \(k\) ordered groups
\(\pi_i= N_i /\sum_i N_i\)
\(\mu_i\) is the relative mean income of group \(i\)

\[ P_{ER} (\alpha) = \sum_{i=1}^n \sum_{j=1}^n \pi_{i}^{\alpha+1} \pi_j \left| \mu_i - \mu_j \right| \]

The parameter \(\alpha\) is the degree of sensitivity to polarization. When \(\alpha=0\), \(P_{ER}(\alpha) = 2 G_{B}\), where \(G_B\) is the Gini index between groups.

Spatial Polarization (Zhang and Kanbur 2001)

\[ I(y) = \sum_{g}^K w_g I_g + I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k) \] \[ w_g = \begin{cases} f_g \left( \frac{\mu_g}{\mu} \right)^c & \text{if } c \neq 0, 1 \\ f_g \left( \frac{\mu_g}{\mu} \right) & \text{if } c = 1 \\ f_g & \text{if } c = 0. \end{cases} \] \(I_g\) is the inequality in group \(g\), \(\mu_g\) is the mean of incomes for members of group \(g\) and \(e_g\) is a unity vector with \(n_g\) elements, and \(f_g=\frac{n_g}{n}\). The parameter \(c\) measures transfer sensitivity. For \(c < 2\), the measure is more sensitive to transfers in the bottom of the distribution than to those in the top end of the distribution. When \(c=1\) or \(c=0\) the Theil (1972) measure applies.

Spatial Polarization

The measure of spatial polarization is the ratio of the between-group inequality over the within-group inequality: \[ P_{ZK} = \frac{ I(\mu_1 e_1, \mu_2 e_2, \ldots, \mu_k e_k)}{\sum_{g}^K w_g I_g }. \]

The polarization can thus grow with increasing distances between the average incomes of the regions as well as a decrease in the inequality within the regions.

Inspiration

Bipolarization

Attribute Bipolarization (Wolfson 1994)
- Attribute distribution used to define groups
Spatial Polarization (Esteban and Ray 1994; Zhang and Kanbur 2001)
- Regionalization used to define groups
- Interpret alienation polarization as interregional inequality
- Interpret identification polarization as intraregional inequality

Integration

Combine attribute and spatial polarization
Embed in a dynamic framework à la Quah (1996)

Family of Spatial Polarization Measures

Spatial Scale

identification of polarization clusters
heirarchical scales of polarization (global-meso-local)

Dynamics

relax fixed regionalization assumption
extend traditional interregional inequality decomposition framework

Framework

Polarization as Graph Intersection

Areas as nodes

Each area will be a vertex \(v_i\), \(i=1,2,\ldots, n\)
\(|V| = n\)

Two Graphs

Distributional (Attribute) Polarization Graph
Geographical (Spatial) Graph

Attribute Polarization Graph

\[A = (V_A, E_A)\]

\[E_A = \{(i, j) \in V \times V \mid g(f(i)) = g(f(j))\}\] where \(g(f(i))\) is a function mapping \(i\) to a value class. The specific form of this mapping obtains from the type of polarization under consideration.

Geographical Adjacency Graph

\[G = (V_G, E_G)\] codifies the geographical relationships between each pair of locations.

Components

Both graphs share identical vertex sets, \(V_A = V_G = V\), with each vertex linked to a locational observation \(i\), and \(|V| = n\).

Let \(C_G = \{C_{G,1}, C_{G,2}, ..., C_{G,k_{G}}\}\) define the \(k_G\) connected components of \(G\).
Similarly, define \(C_A = \{C_{A,1}, C_{A,2}, ..., C_{A,k_{A}}\}\) as the component set for \(A\).
Let \(k=max(k_G, k_A)\).

Intersection Graph

\[I=(V_I, E_I)\] where

\(V_I=V_G=V_A\)
\(E_I = E_G \cap E_A\)

Define \(C_I = \{C_{I,1}, C_{I,2}, \ldots, C_{I,k_{I}}\}\) as the component set for \(I\), with \(k_{I} = \left | C_I \right |\).

Spatial Polarization Index

The index of spatial polarization is: \[ S(G, A) = 1 - \frac{k_I - k}{ (n - k)} \]

\(k =max(k_G, k_A)\).

Spatial Bipolarization

Given a set of \(n\) locations with a vector of per capita incomes \(Y\), let \(m= \text{median}(Y)\). Define \(A\) as the \(n \times n\) attribute adjacency graph with edge conditions: \[ A_{ij} = \begin{cases} 1 & \text{if } (Y_i \leq m \text{ and } Y_j \leq m) \text{ or } (Y_i > m \text{ and } Y_j >m), \\ 0 & \text{otherwise}. \end{cases} \]

\(A\) will have \(k_A=2\) components of equal size when \(n\) is even, and approximately equal size when \(n\) is odd.

Define \(G\) as the spatial adjacency graph with: \[ G_{ij} = \begin{cases} 1 & \text{if observations } i \text{ and } j \text{ are geographically adjacent}, \\ 0 & \text{otherwise}. \end{cases} \] The number of connected components for the spatial adjacency graph, \(k_G\), depends on the locational configuration. In a connected spatial adjacency graph, there is a path between each pair of locations, and thus there will be no isolated vertices or disconnected subgraphs. In such a graph \(k_G=1\).

Spatial Bipolarization

\[\begin{equation} S^B(A, G) = 1 - \frac{k_I-k}{n-2}. \label{eq:sp} \end{equation}\] with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).

\[ 0 \le S^B \le 1\]

\[ S^B = \begin{cases} 1 &k_I = 2 \\ 0 &k_I = n. \end{cases} \]

Spatial Bipolarization

Views

Spatial distribution of attribute polarization
Polarization of the spatial join structure

Scales of Spatial Bipolarization

\[ V = C_{I,1} \cup C_{I,2} \cup \dots \cup C_{I,k_I} \] with the property that:

\[C_{I,f} \cap C_{I,h} = \emptyset \ \forall \ f \ne h \]

Spatial Polarization Clusters

\[C_{I,f}\]

a path exists between each pair of member nodes in the spatial adjacency subgraph induced by the intersection
a path exists between each pair of member nodes in the attribute graph

Scales of Spatial Bipolarization

Global

\[S^B\]

Meso

\[C_I\]

Local

\[ C_{I,l} = \{ V_j \in V : V_j \text{ is in component } l \} \]

Meso Scale: \(C_{I,f}\)

For each component \(f\) of graph \(I\), consider the measure:

\[\begin{equation} \zeta_f = \frac{\sum_{i \in f} \sum_j G_{i,j} A_{i,j}}{\sum_{i \in f} \sum_j G_{i,j}}. \end{equation}\]

For the nodes in this intersection component, this represents the proportion of incident edges from the geographical adjacency graph that also exist in the intersection component. In other words, this is the share of each location’s geographical neighbors that belong to the same attribute component as the location.

Local Scale: \(V_i\)

\[ \zeta_i =\frac{\sum_j G_{i,j}A_{i,j}} {\sum_j G_{i,j}} \]

Global-Meso-Local

Global

\[ \zeta = \frac{\sum_i \sum_j G_{i,j} A_{i,j}}{\sum_i \sum_j G_{i,j}} \]

Global from Meso

\[ \zeta = \sum_f \frac{\sum_{i \in f} \sum_j G_{i,j}} {\sum_i \sum_j G_{i,j}} \zeta_f \]

Global from Local

\[\zeta = \sum_i \frac{\sum_j G_{i,j}}{\sum_i \sum_j G_{i,j}}\zeta_i\]

Dual Hierarchy for Attribute Polarization

\[ \nu_f = \frac{\sum_{i \in f} \sum_j G_{i,j} A_{i,j}}{\sum_{i \in f} \sum_j A_{i,j}} \]

and

\[ \nu_i = \frac{\sum_j G_{i,j} A_{i,j}} {\sum_j A_{i,j}} \]

Spatial Multipolarization

Quartile Classification

In a quartile-based grouping there would be four components in the attribute graph and, assuming the spatial graph has a single component, our index of spatial multipolarization becomes: \[ S^M(A, G) = 1 - \frac{k_I-4}{n-4}. \]

The reliance on quartiles for the attribute grouping gives an interesting cartographic interpretation to the multipolarization measure. The index is inversely related to the number of components in the intersection graph \(k_I\). Each of these components is a face in the choropleth map. In the cartographic literature a common measure of map complexity is the face ratio, defined as the number of observed faces over the number of faces in the original map (cite: MacEachren 1982). The number of observed faces is the number of faces that obtain when internal edges separating areas assigned to the same attribute class are dissolved. The measure of spatial multipolarization is a slightly modified complement to this measure of map complexity, with adjustments made for the relative number of components in the spatial and attribute graphs.

Map Complexity (MacEachren 1982)

High and Low Map Complexity

Empirical Illustration

Mexico

States

Trends

Distributions

Spatial Dynamics

Relative Quartiles

Queen Adjacency Graph

Bipolar Adjacency Graph 1940

Spatial Bipolarization Graph

Spatial Bipolarization

\[\begin{equation} S^B(A, G) = 1 - \frac{k_I-k}{n-2} = 1-\frac{6-2}{32-2} = 0.87. \label{e:S2} \end{equation}\]

with \(k_I = |C_I|\) , \(I = A \cap G\), and \(k = max(k_A, k_G)\).

Dynamics of Inequality and Spatial Polarization

Scales of Spatial Polarization

\[\begin{equation} 1 - \frac{k_{i,low} - 1}{n_{low}-1} = 1- \frac{1_{} - 1}{16-1} = 1 \end{equation}\] and for the high end of the distribution we have: \[\begin{equation} 1- \frac{k_{i,high} - 1}{n_{high}-1} = 1 - \frac{5 - 1}{16-1} = 0.73 \end{equation}\] Note that the overall spatial polarization index is the average of these two components.

Local neighborhood homogeneity index Mexico state income 1940.

Local class connectivity index Mexico state income 1940.

The distributional corollary to this index is the measure of local class connectivity from equation (\(\ref{e:locala}\)) shown in Figure [[fig:locala]], which measures the proportion of a state’s attribute neighbors that are also its geographical neighbors.[fn:4]

The two sets of local measures are complements. The former measure tells us the extent of local spatial polarization experienced by state \(i\), while the latter measure informs on the relative contribution of state \(i\) to overall attribute polarization. We can see the complementarity by contrasting the states of Baja California Sur and Oaxaca. Both of these states have high values for their local neighborhood indices, however, Baja California Sur has a low value for the local class connectivity measure while Oaxaca has a higher value for this index reflecting the spatial fragmentation (concentration) of the high (low) income components of the attribute distribution and the respective roles of the two states in these components.

Taken together, the three scales of spatial polarization measured by the indices address three distinct questions. From the global perspective, is the attribute polarization concentrated in space? From the meso perspective, which parts of the attribute distribution drive the spatial polarization - or is there a distinction between the spatial articulation of the high and low ends of the distribution? And, from the local perspective we can ask which individual states contribute to, and/or experience, the spatial polarization?

Spatial multipolarization graph 1940 (quartiles: Q1 dark blue, Q2 blue, Q3 red, Q4 dark red).

Figure [[fig:multigraph]] portrays the graph induced by the intersection of the quartile attribute and queen contiguity graphs. This multipolarization graph has 15 connected components, resulting in a polarization index of 0.61. The more granular slicing of the attribute distribution into quartiles further emphasizes the asymmetry in the spatial income distribution as the largest two components arise from states falling in the bottom two classes of the income distribution - all the states with incomes below the first quartile comprise the biggest component with eight states (dark blue). The next largest component comprises seven states belonging to the second income class. The isolate Campeche belongs to the second class. In other words, subdividing the bottom half of the income distribution into two attribute components yields only three spatial components in the multipolarization intersection graph. By contrast, the two income classes above the median are highly fragmented into five and seven components.

Regimes

Hanson (1996)

Spatial bipolarization intersection graph using the regionalization scheme from Hanson (1996).

This difference suggests two different perspectives on what the polarization index captures when applied to the block graph case. On the one hand, it is measuring the extent to which “neighboring” pairs of observations belong to the same side of the income distribution. This is in the spirit of the Wolfson polarization index. On the other hand, the definition of neighbor relies on regional membership rather than spatial contiguity. In other words, all pairs of observations assigned to the same region contribute equally to the index, whether or not the pair of locations is spatially contiguous. When groups of spatially connected observations from the same region are on different sides of the median, what was one face in the map of the regional partition will become multiple faces in the map of the intersection component. Because of the difference between the number of components in the intersection graph and the number of faces in the map of these components, we might view the polarization index from the intersection graph as a measure of regional polarization rather than spatial polarization. To have an index of spatial polarization for the block neighbors case, we could add a subsequent intersection of the block-attribute graph with the spatial contiguity graph. The induced graph incudes 14 components which now are identical to the faces in the map of Figure [[fig:hansoncomponents]]. Using the number of spatial components in the polarization index drops the value to 0.69.[fn:2]

Polarization as Regional Inequality Decomposition

Conclusion

Contributions

Family of spatial polarization indices
Graph based frameing
Distinguish between concentrated advantage and concentrated disadvantage
Cluster identification
Dynamic spatial partitions

Next Steps

Comparative spatial polarization (US-EU, US-Mexico, etc.)
Properties of polarization index
Multivariate extensions
pysal-inequality

Thanks

References

Esteban, J. M, and D. Ray. 1994. “On the Measurement of Polarization.” Econometrica 62 (4): 819–51.

Hanson, Gordon H. 1996. U.S.-Mexico Integration and Regional Economies. Cambridge: National Bureau of Economic Research.

Kaplan, Ethan, Jörg L. Spenkuch, and Rebecca Sullivan. 2022. “Partisan Spatial Sorting in the United States: A Theoretical and Empirical Overview.” Journal of Public Economics 211 (July): 104668. https://doi.org/10.1016/j.jpubeco.2022.104668.

MacEachren, Alan M. 1982. “Map Complexity: Comparison and Measurement.” The American Cartographer 9 (1): 31–46. https://doi.org/10.1559/152304082783948286.

Pike, Andy, Vincent Béal, Nicolas Cauchi-Duval, Rachel Franklin, Nadir Kinossian, Thilo Lang, Tim Leibert, et al. 2024. “‘Left Behind Places’: A Geographical Etymology.” Regional Studies 58 (6): 1167–79. https://doi.org/10.1080/00343404.2023.2167972.

Quah, Danny T. 1996. “Twin Peaks: Growth and Convergence in Models of Distribution Dynamics.” The Economic Journal 106 (437): 1045–55. https://doi.org/10.2307/2235377.

Rey, Sergio J., and Esau Casimiro Vieyra. 2023. “Spatial Inequality and Place Mobility in Mexico: 2000–2015.” Applied Geography 152 (March): 102871. https://doi.org/10.1016/j.apgeog.2023.102871.

Rodríguez-Pose, Andrés. 2018. “The Revenge of the Places That Don’t Matter (and What to Do about It).” Cambridge Journal of Regions, Economy and Society 11 (1): 189–209. https://doi.org/10.1093/cjres/rsx024.

Theil, Henri. 1972. Statistical Decomposition Analysis: With Applications in the Social and Administrative Sciences. Studies in Mathematical and Managerial Economics 14. Amsterdam: North-Holland Publ. [u.a.].

Wolfson, Michael C. 1994. “When Inequalities Diverge.” The American Economic Review 84 (2): 353–58. https://www.jstor.org/stable/2117858.

Zhang, Xiaobo, and Ravi Kanbur. 2001. “What Difference Do Polarisation Measures Make?: An Application to China.” Journal of Development Studies 37: 85–98.